[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572419-105":53,"doc-detail-572419-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","mathematics-differential-equations-and-laplace-transforms","Mathematics - Differential Equations and Laplace Transforms","","This document provides a concise overview of core concepts in differential equations and transform methods. It covers essential formulas for Laplace transforms, including definitions and shifting theorems, followed by a systematic guide to first-order differential equations, specifically separable equations and linear forms. Additionally, it details methods for second-order linear differential equations with constant coefficients and introduces stability concepts for systems of ordinary differential equations through eigenvalue analysis in the complex plane.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/mathematics-differential-equations-and-laplace-transforms/572419/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/mathematics-differential-equations-and-laplace-transforms/572419.png","ImageObject",442,249,{"name":88,"@type":89},"SundayMiles","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What are the conditions for system stability based on eigenvalues?","Question",{"text":108,"@type":109},"If all poles (eigenvalues) are on the left of the complex plane, the system is asymptotically stable. If at least one pole is on the imaginary axis, it is neutrally stable, and if at least one pole is on the right side of the imaginary axis, the system is unstable.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How is a Bernoulli equation transformed into a linear differential equation?",{"text":113,"@type":109},"A Bernoulli equation of the form y' + p(x)y = r(x)[y]^n is transformed by using the substitution z(x) = [y(x)]^(1-n).",{"name":115,"@type":106,"acceptedAnswer":116},"What is the Laplace transform of a derivative?",{"text":117,"@type":109},"The Laplace transform of the first derivative is L{f'} = sL(f) - f(0), and for the second derivative, it is L{f''} = s^2L(f) - sf(0) - f'(0).","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572419,1792546172,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":76},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Table of derivatives and integrals  \nFirst order DEs  \nSeparable equations  \nSome special cases  \nEquation y􀁣 = f  \nEquation y􀁣 = f  \n􀂧  y  􀂷  y  \n􀂨􀂩 x 􀂹􀂸 is transformed to separable with substitution u = x 􀂟 y = ux  \n(ax+by+k ) is transformed to separable with substitution u = ax+by+ k  \nFirst order linear DEs  \nEquation y 􀁣 (x) + p(x)y (x) = r (x) is transformed to separable using the integrating factor 􀁕(x) = e􀂳 p ( x)dx  \nBernoulli equation y 􀁣 (x) + p(x)y (x) = r (x)[ y(x)]􀁄 (􀁄 􀁺1&􀁄 􀁺 0) is transformed to linear using substitution z(x) = [ y(x)]1−􀁄  \nSecond order linear differential equation with constant coefficients  \ny 􀁣􀁣 + ay􀁣 + by = r (x)  \na, b − const.  \nStructure of general solution. General solution a sum of general solution of homogeneous equation and particular solution of the nonhomogeneous equation.  \nHomogeneous case  \ny 􀁣􀁣 + ay􀁣 + by = 0 (1)  \nSolution is determined by solving the corresponding characteristic equations  \n􀁏2 + a􀁏 + b = 0 (2)  \nParticular solution of the non-homogeneous equation  \ny 􀁣􀁣 + ay􀁣 + by = r (x)  \n(4)  \na, b − const.  \nSystems of ODEs (planar case, critical/stationary points, stability)  \nLaplace transform method  \nLinear systems of ODEs with Laplace transform","cbCaiaWuXZnFjPoF","https://ap.wps.com/l/cbCaiaWuXZnFjPoF","pdf",921178,9,"English","# Laplace Transforms\n## Definition and Properties\n# Differential Equations\n## First Order Equations\n## Second Order Linear Equations\n# System Stability\n## Eigenvalue Analysis","[{\"question\":\"What are the conditions for system stability based on eigenvalues?\",\"answer\":\"If all poles (eigenvalues) are on the left of the complex plane, the system is asymptotically stable. If at least one pole is on the imaginary axis, it is neutrally stable, and if at least one pole is on the right side of the imaginary axis, the system is unstable.\"},{\"question\":\"How is a Bernoulli equation transformed into a linear differential equation?\",\"answer\":\"A Bernoulli equation of the form y' + p(x)y = r(x)[y]^n is transformed by using the substitution z(x) = [y(x)]^(1-n).\"},{\"question\":\"What is the Laplace transform of a derivative?\",\"answer\":\"The Laplace transform of the first derivative is L{f'} = sL(f) - f(0), and for the second derivative, it is L{f''} = s^2L(f) - sf(0) - f'(0).\"}]","Mathematics - Differential Equations and Laplace Transforms | PDF",1791927314]